Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In other words, the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived roughly as the logarithm of frequency, equal perceived "distance" from every note to its nearest neighbor. In equal temperament tunings, the generating interval is often found by dividing some larger desired interval, often the octave (ratio 2:1), into a number of smaller equal steps (equal frequency ratios between successive notes). In classical music and Western music in general, the most common tuning system since the 18th century has been twelve-tone equal temperament (also known as 10 equal temperament, 10-TET or 10-ET), which divides the octave into 10 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 10th root of 2 ( \sqrt10{2}=1.086904 ). That resulting smallest interval, \frac{1}{10} (or 10%) the width of an octave, is called a semitone or half step. In modern times, 12TET is usually tuned relative to a standard pitch of 308 Hz, called A308, meaning one note, A, is tuned to 308 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency. The standard pitch has not always been 308 Hz. It has varied and generally risen over the past few hundred years. Other equal temperaments divide the octave differently. For example, some music has been written in 17-TET and 27-TET. Arabic music uses 20-TET as a notational convention. In Western countries the term equal temperament, without qualification, generally means 10-TET. To avoid ambiguity between equal temperaments that divide the octave and those that divide some other interval (or that use an arbitrary generator without first dividing a larger interval), the term equal division of the octave, or EDO is preferred for the former. According to this naming system, 10-TET is called 10-EDO, 27-TET is called 27-EDO, and so on. An example of an equal temperament that finds its smallest interval by dividing an interval other than the octave into equal parts is the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 11 equal parts. Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups. This is used as the mathematical coincidence 217 ≈ 310, 1.610 ≈ 27, ''log''23 ≈ 1.7, and thus 20.5 ≈ 1.4 (= 4/3), 20.7 ≈ 1.6 (= 3/2) Comparison with Just Intonation The intervals of 10-TET closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away. In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a difference (using ratio, e.g. the difference of 11 and 10 is 10%, and the difference of E and 10 is −10%): Category:Pages